Characterization of Continuity and Convergence of Fredholm Operators Perturbed by Orthogonal Idempotents in Banach Spaces - Kiprono, Nyaluke Wesley (JOOUST, 2021)
Fredholm operators have been studied by several researchers for decades whereby they developed a comprehensive theory for integral equations and integral operators. A lot of properties of Fredholm operators have been given considerations and interesting results have been obtained. For instance, Fredholmness, continuity, boundedness, compactness, and linearity have been given consideration. However, continuity and convergence of Fredholm operators have been considered but without perturbation. This study therefore seeks to characterize Fredholm operators in terms of continuity and convergence when perturbed by orthogonal idempotents. The objectives of the study were to: determine conditions under which Fredholm operator retains Fredholmness when perturbed by orthogonal idempotents in Banach spaces; characterize continuity of Fredholm operators perturbed by orthogonal idempotent in Banach spaces; and characterize convergence of Fredholm operators perturbed by orthogonal idempotent in Banach spaces. The methodology involved tensor products, direct sum decomposition, spectral decomposition and other fundamental principles. The results obtained show that; Fredholm operators perturbed by orthogonal idempotents are continuous if FOI (H) is three dimensional and for J1, J2 ∈ IO(H), T ∈ FOI (H) is continuous and J1 ∼ J2 if and only if ∃ C, D ∈ A ⊆ FOI (H) such that C 6= D, J1 ≤ C, J1 ≤ D, J2 ≤ C and J2 ≤ D. In addition, A ⊆ F c OI (H) then T ∈ F c OI (H) is compact if for a sequence xn ∈ A, xn * x and xn − T xn → 0, x is a fixed point of T.